Question

Solve: ${\int }_0^a\frac{dx}{\sqrt{ax-x^2}}$

Answer 1

${\int }_0^a\frac{dx}{\sqrt{ax-x^2}}$

$={\int }_0^a\frac{dx}{\sqrt{\frac{a^2}{4}-\frac{a^2}{4}+2\times \frac{a}{2}x-x^2}}$

$={\int }_0^a\frac{dx}{\sqrt{\frac{a^2}{4}-(x-\frac{a}{2}{)}^2}}$

$={\sin }^{-1}\left(\frac{x-\frac{a}{2}}{\frac{a}{2}}\right){\int }_0^a$

$={\sin }^{-1}\left(\frac{a-\frac{a}{2}}{\frac{a}{2}}\right)-{\sin }^{-1}\left(\frac{0-\frac{a}{2}}{\frac{a}{2}}\right)$

$=\frac{\pi }{2}-(-\frac{\pi }{2})$

$=\pi$.